No hostages were seriously hurt but the dead woman was apparently a bystander caught in crossfire between the gunman and police on Saturday afternoon.

About two hours before taking the hostages, police say the man shot his grandmother seven times and wounded another woman, who he forced into a car.

Police chased the vehicle and exchanged gunfire with the man, who crashed into a pole outside the Trader Joe's in the city's Silver Lake section and ran inside.

Frightened customers and workers were forced to dive for cover as police bullets fired at the suspect shattered the store's glass doors. Some inside the supermarket climbed out windows and others barricaded themselves in rooms as scores of police and firefighters and 18 ambulances converged on the scene and prepared for mass casualties.

Heavily armed officers in riot gear stood along the side of the store and used mirrors to look inside as hostage negotiators tried to coax the man into freeing his 40-to-50 hostages and surrendering.

At about 6:30pm the man agreed to handcuff himself and walked out the front door, surrounded by four of the hostages. The unidentified man, who police said is about 28, was immediately taken into custody. Police said he had a wound to his arm.

Mayor Eric Garcetti congratulated police and firefighters for their work and mourned the loss of life at the Trader Joe's.

"The heroism that was shown today was second to none and the teams that were able to respond, secure the perimeter and engage in conversation with the suspect no doubt saved lives today," he said, adding "our hearts go out to everyone who has been traumatised."

Police Chief Michael Moore said the suspect made a "series of demands" during the standoff but crisis negotiators believed they could convince him to surrender peacefully.

Police aren't sure what led to the initial violence that produced the car chase and standoff. Moore said at about 1:30pm the suspect shot his grandmother and another woman in a South Los Angeles home and then forced the other woman into his grandmother's car. The grandmother was in critical condition while the other woman suffered a grazing wound.

Officers were able to track the car using LoJack - a stolen vehicle tracking system - and officers tried to stop the suspect in Hollywood, but the man refused to pull over, Moore said. During the chase, the suspect fired at officers, shooting out the back window of his car.

Outside the store, the man exchanged gunfire with police again where the woman was killed, Moore said.

It was not clear if she died from police gunfire or was killed by the gunman. Moore said police and firefighters responded quickly but could not save her.

Fire officials said six people, ranging in age from 12 to 81, were taken to the hospital. None had been shot and all were in fair condition.

Australian Associated Press